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\usepackage{amsmath, amssymb} % 数学公式与符号
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\title{《复变函数》课程论文选题}
\author{五六七}

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\begin{document}
\maketitle 

\begin{enumerate}

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%\section{课文三}
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\item % 1
球极投影将球面上的任意一个圆周映照为平面上的一个圆周或一条直线。

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高斯-卢卡斯定理。

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Smale平均值猜想。

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有理函数的图像。Riemann-Hurwitz定理。

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多项式的图像。

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多项式的图像与幂级数的图像的本质区别。

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\item % 1
指数函数的图像。对数函数的解析单值分支。

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%\section{课文四}
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\item % 1
分式线性变换将圆周和直线变为圆周和直线。

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\item % 1
（对称原理）分式线性变换保持对称性。

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分式线性变换 $w=\frac{az+b}{cz+d}$ 的图像（复平面）。

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分式线性变换 $w=\frac{az+b}{cz+d}$ 的图像（黎曼球面）。

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\item % 1
有理函数 $w=\frac{1}{2}\left( z+\frac{1}{z} \right)$ 的图像。

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二次多项式的图像。

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三次多项式的图像。

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正弦函数的图像。反正弦函数的解析单值分支。

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余弦函数的图像。反余弦函数的解析单值分支。

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\item % 1
正切函数的图像。反正切函数的解析单值分支。

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\item % 1
将两个相交的圆盘的公共区域映为半平面。

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将相切的两个圆盘之间的区域映照为半平面。

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例子4.2. 求将右半平面除去 $[0,1]\cup [x,\infty)\, (x>1)$ 映为半平面的共形映射。

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\item % 1
例子4.3. 求将半平面映为双曲线 $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 的右半分支左边部分的共形映射。

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%\section{习题四}
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%\begin{enumerate}
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\item % 1

求将 $0, i, -i$ 变为 $1, -1, 0$ 的分式线性变换。
    

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\item % 2

如果一个四边形的顶点依次为 $z_1, z_2, z_3, z_4$, 且都位于一个圆周上，证明
$$
|z_1 - z_3| \cdot |z_2 - z_4| = |z_1 - z_2| \cdot |z_3 - z_4| + |z_2 - z_3| \cdot |z_1 - z_4|.
$$

并给出几何解释。
    

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\item % 3

证明反射将圆周变为圆周。
    

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\item % 4

试求将虚轴、直线 $x = y$、单位圆周分别为圆周 $|z-2|=1$ 的反射。
    

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\item % 5

求将圆周 $|z|=2$ 变为 $|z+1|=1$、将点 $-2$ 变为原点、将原点变为 $i$ 的分式线性变换。
    

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\item % 6

假定一个分式线性变换将一对同心圆周变为另一对同心圆周，证明圆周半径之比不变。
    

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\item % 7

求把单位圆周与圆周 $|z-1/4|=1/4$ 变为同心圆周的分式线性变换。
    

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\item % 8

求如下分式线性变换的不动点，并判断它们的类型:

(1) $\dfrac{z}{2z-1}$

(2) $\dfrac{2z}{3z-1}$

(3) $\dfrac{3z-4}{z-1}$

(4) $\dfrac{z}{2-z}$
    

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\item % 9

证明 $n (n \geq 2)$ 次复合为恒等映射的分式线性变换一定是椭圆变换。
    

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\item % 10

求所有与单位圆周以及 $|z-1|=4$ 正交的圆周。
    

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求将单位圆盘与圆盘 $|z-1|<1$ 的公共部分映为单位圆盘的共形映射。
    

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\item % 12

求将单位圆周与圆周 $|z-1/2|=1/2$ 之间的区域映为半平面的共形映射。
    

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\item % 13

求将圆弧 $\{z=x+iy: |z|=1, y \geq 0\}$ 的补区域映为单位圆盘外部，并保持无穷远点不动的共形映射。
    

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\item % 14

求将抛物线 $y^2=2px$ 的外部映为单位圆盘，且把 $z=0$ 及 $z=-p/2$ 分别变为 $w=1$ 及 $w=0$ 的共形映射。
    

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\item % 15

求将双曲线 $x^2-y^2=a^2$ 的右半分支的右边部分映为单位圆盘，且把焦点变为 $w=0$, 把顶点变为 $w=-1$ 的共形映射。
    

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\item % 16

求将椭圆 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ 的外部映为单位圆盘的共形映射。

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%第7-11章
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\item % 1
将一个三角形映照为半平面。

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\item % 1
将一个四边形映照为半平面。

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\item % 1
迪利克雷问题，泊松公式。

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\end{enumerate}

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\end{document}

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